Ergodic perturbations of degenerate integrable Hamiltonian systems

Hamiltonian systems arbitrarily close in the $C^r$ topology $(r=1,2,\dots)$ to a given integrable degenerate Hamiltonian system of class $C^\infty$ which generate an ergodic flow on each manifold of constant energy are constructed. Applications: Small perturbations of a system generated by independent oscillators and Finsler metrics close to standard Riemannian metrics on symmetric spaces of rank 1.